I think there is a fix actually: go to the graveyard and find a murdered woman, I can't remember her name, though a guard is there. You must tell the guard you are willing to help to get the murderer, then you will start a quest that involves research in the house. Then you can buy the house.

The problem I'm having is that there is no quest to complete. :( No MISC quest added, yet I've used all the dialog options in town for the 'people talking about murders'. And if by graveyard you mean the place by Hall of Dead, it is empty for me unless the priest is praying(whom has no dialog about murders).
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JClaspillNov 25 '11 at 8:44

yeah, ive been in the house got all the journals, the strange amulet that was hidden in there and collected the pamphlets. There was never a body in the grave yard and the quest never begun. i can talk about the house but not buy it! i only went into the house and collected the things to see if that would help the quest to start for me as the body just wouldnt appear... its very annoying :S the quest you should get is call "blood on the ice"
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user15315Nov 26 '11 at 2:04

You need to start or complete the quest Blood on the Ice (In order to purchase the upgrades for the house the quest must be completed.).

Blood on the Ice is a quest initiated in Windhelm. In order to start it you must have entered and exited Windhelm at least three times.

Also the quest is automatically gained when entering the house (Hjerim) for the first time, if not previously completed.

Walkthrough

Approach the graveyard outside the Hall of the Dead at night; there
will be several people gathered around a female corpse.

Speak with the guard.

Visit Jorleif to be granted permission to help the investigation.

Return to the guard.

Enter the Hall of the Dead and speak with Helgird.

Return to the body, follow the trail of blood to Hjerim.

Return to Jorleif for advice.

Pick the lock or receive a key from the Tova Shatter-Shield, Friga
Shatter-Shield's mother and enter the house.

Investigate all of the furniture that gives you the option.The
wardrobe on the north east side from entering the house will have a
hidden door leading to an altar.
10.Take the Butcher's Journal #2, exit and grab the first journal from
the bloody chest up against the wall, Strange Amulet and flier from
the shelf (if you turn around and face the main door, the amulet is
on the shelves on the right side of the door, under the fliers).

Visit the Candlehearth Hall or Calixto's House of Curiosities to
speak with Calixto Corrium. You can show him the Strange Amulet and
even sell it to him, if you so choose. He will tell you it belongs
to the court mage.

Visit Viola Giordano. She's often found in Candlehearth Hall (note:
you can't talk to her if you don't have a copy of the "Beware the
Butcher" pamphlet in your inventory).

At this point, the player can complete this quest in one of two ways:

Accuse the mage.
Return to Jorlief to end the quest and thus accuse the court mage of the murders.

or

Catch the killer

Go and talk to Wuunferth, the court mage, rather than following the
quest marker to Jorleif. Tell him you heard he was practising
necromancy. He tells you he doesn't, and that the amulet and
journals are not his.

Go to the marketplace area in the stone quarter at night, as
Wuunferth suggests, find and stop Calixto, who is about to strike
again.

Return to Jorleif and tell him that the killer has been caught.

When completing the quest, talking to the Steward allows the Dragonborn to buy upgrades for Hjerim. Upon becoming Thane of Eastmarch, Calder is pronounced housecarl to the Dragonborn and lodges at Hjerim.

Aditional Notes to the Quest and the House

Purchasing the Clean-up upgrade is not necessary, but it will be
unavailable after purchasing the Living Room upgrade.

Although the Windhelm Home Decorating Guide lists a Guest Room, it
cannot be purchased; it is automatically installed.

In order to purchase Hjerim, you must have either completed the
Rescue From Fort Neugrad quest if you've joined the Stormcloaks.

If Falkreath was given to the Stormcloaks in Season Unending before
you received the Rescue From Fort Neugrad quest in the Stormcloaks
Civil War questline or you completed the Civil War quest line as part
of the Legion,then you must complete Blood on the Ice in order to
purchase the House.

If you do not sell the amulet to Calixto and kill him as part of Blood on the Ice, it will be stuck in your inventory as a quest item. You can loot it from his body after you kill him as part of the quest.

Starting "Blood on the Ice"

You need to finish the Blood on the Ice quest first before the steward will give you the conversation option to buy the Windhelm house (Hjerim). Make sure you meet the requirements needed to start that quest:

"This quest is triggered by a counter that starts at the beginning of
the game which will increase each time you visit the city of Windhelm
or the immediate surrounding area. When it reaches four, the Blood on
the Ice quest will be available, although two more conditions must be
met:

The player has to enter Windhelm, either through the main gate, the docks gate, or by fast-traveling, between 7pm and 7am. The graveyard
scene should then be triggered and the quest ready to go.

The easiest way to trigger the quest is to walk back and forth between
Windhelm Stables and Brandy-Mug Farm two-three times, then
enter Windhelm between 7pm and 7am. If the graveyard scene is not
triggered by this, there's little hope you can trigger the quest at
all.

NOTE: You can always check whether or not you have met the
requirements for the quest to start, as the lock to Hjerim will change
from a “key required”-lock to a normal lock when the quest becomes
available.

If Tova and Arivanya are alive, the quest has not yet initiated
because the counter is less than four, and you obtain a key to Hjerim,
then you can trigger this quest by entering Hjerim with that key and
picking up one of the Beware the Butcher! pamphlets. However, the
quest will not start in the same way as outlined in the walkthrough
above; you will have to figure out what to do on your own without help
from Jorleif, however all of the same options will be open to you
except that none of the witnesses will be present in the graveyard;
you can sell Calixto the Strange Amulet, you can have Wuunferth
arrested or talk to him to catch Calixto, and you will have to talk to
Viola in order to continue on with the quest after picking up a
pamphlet."

Possible conflicts with the Dark Brotherhood quest, "Mourning Never Comes"

Although most of the NPCs needed for the quest to initiate are marked as essential, another quest, namely the Dark Brotherhood quest Mourning Never Comes, will encourage you to kill Nilsine Shatter-Shield. Even though she is not related to this quest, her mother, Tova Shatter-Shield, will commit suicide if Nilsine is killed, and will therefore render this quest impossible to initiate.

If you've killed Nilsine in order to get a bonus reward in the quest Mourning Never Comes, Tova will be dead inside her house, and you will need to pick the lock of her house in order to get the key to Hjerim. This will only happen if you met the requirements for Blood on the Ice and then completed Mourning Never Comes.

Also, if you are playing on the PC, install the Unofficial Skyrim Patch to fix a lot of bugs related to Blood on the Ice quest - one of which is related to the key to Hjerim:

If Tova Shatter-Shield is dead (actually just disabled, but that's another story) because you completed "Mourning Never Comes" (DB03) before MS11, a substitute key for Hjerim will be made available by the USKP.

Already finished "Blood on the Ice"

If you've already finished Blood on the Ice and the steward is still not giving the conversation options for inquiring about or buying the house, then you might have encountered a bug. If you're playing on the PC, the console command setstage 000A7B33 10 should allow the steward to give the conversation options necessary for you to buy the house. (As suggested by the UESP wiki's Hjerim - Notes article.) This worked for one of my characters which has encountered that bug.

My game seriously bugged this quest too, couldn't start it no matter what. But this is what I ended up trying and I was able to successfully play through the quest without any more bugs appearing (that's right, not skipping through it). Afterward I even got an unglitched house (no 'mess').

you will need the console

Walk up to Hjerim and unlock it using the console. You do this by pressing ~, clicking on the door and entering unlock. Hopefully this is the only time you'll need to use the console.

Go inside and head to the opposite end of the house where there is a stack of paper. Take only one of the flyers, this should start the quest with Follow up on the clues from Hjerim. Leave the house and re-enter, this will set the quest stage to Look for clues which puts the quest in a working state!

Read on for completion details of how I got through the rest of the bug ridden quest from this point onward (spoilers).

* Go and find the journals in the chest to the left of the door, and in the secret room behind a closet next to the stack of paper. Before you leave, get the amulet hiding underneath the flyers.

* Go and find Viola Giordano, if you can't you can try the movetoconsole command on her (player.moveto 0001B13C). Tell her about the journal and she will suspect the court mage.

* At this point I would advise making sure Arivanya is alive. You can do this by reviving her, shouldn't do anything averse if she isn't dead (prid 0001B144, enable, moveto player, resurrect, resetAI then wait 24 hours (for AI to work if she was dead).

* Now go to the court mage and he will tell you to stake out the stone quarter at midnight. Go to the marketplace and wait until 10PM, you will witness Calixto killing Arivanya. Kill him on the spot (guards helped me, so there shouldn't be a bounty issue)!

* Done! The quest should tell you to go to Jorleif, which will complete the quest. Buy the house if you can & want, and if you do, immediately decorate with Clean up that murderer's mess or else be at risk of other bugs